I am working with a system whose trajectories converge to the set of equilibria. I can characterize all the equilibria in a nice way and easily compute their stability and whether they are isolated or not. The dynamical system is described by a Lipschitz continuous vector field.
I conjecture that the system is "well behaved" so that the trajectories converge (almost always, depending on their initial condition) to the set of locally stable equilibria. I know that this is not always the case as there are plenty of counterexamples, e.g. this classic system with a homoclinic trajectory (example).
I am just wondering if there is any approach/result available in the literature to show that this is true, given certain conditions on the vector field.